Reading 1, 1 Samuel 1:9-20
9 When they had finished eating in the room, Hannah got up and stood before the Lord. Eli the priest was sitting on his seat by the doorpost of the temple of the Lord.
10 In the bitterness of her soul she prayed with many tears,
11 and she made this vow, ‘Lord of Hosts! Should you condescend to notice the humiliation of your servant and keep her in mind instead of disregarding your servant, and give her a boy, I will give him to the Lord for the whole of his life and no razor shall ever touch his head.’
12 While she went on praying to the Lord, Eli was watching her mouth,
13 for Hannah was speaking under her breath; her lips were moving but her voice could not be heard, and Eli thought that she was drunk.
14 Eli said, ‘How much longer are you going to stay drunk? Get rid of your wine.’
15 ‘No, my lord,’ Hannah replied, ‘I am a woman in great trouble; I have not been drinking wine or strong drink — I am pouring out my soul before the Lord.
16 Do not take your servant for a worthless woman; all this time I have been speaking from the depth of my grief and my resentment.’
17 Eli then replied, ‘Go in peace, and may the God of Israel grant what you have asked of him.’
18 To which she said, ‘May your servant find favour in your sight.’ With that, the woman went away; she began eating and was dejected no longer.
19 They got up early in the morning and, after worshipping the Lord, set out and went home to Ramah. Elkanah lay with his wife Hannah, and the Lord remembered her.
20 Hannah conceived and, in due course, gave birth to a son, whom she named Samuel, ‘since’, she said, ‘I asked the Lord for him.’
Gospel, Mark 1:21-28
21 They went as far as Capernaum, and at once on the Sabbath he went into the synagogue and began to teach.
22 And his teaching made a deep impression on them because, unlike the scribes, he taught them with authority.
23 And at once in their synagogue there was a man with an unclean spirit, and he shouted,
24 ‘What do you want with us, Jesus of Nazareth? Have you come to destroy us? I know who you are: the Holy One of God.’
25 But Jesus rebuked it saying, ‘Be quiet! Come out of him!’
26 And the unclean spirit threw the man into convulsions and with a loud cry went out of him.
27 The people were so astonished that they started asking one another what it all meant, saying, ‘Here is a teaching that is new, and with authority behind it: he gives orders even to unclean spirits and they obey him.’
28 And his reputation at once spread everywhere, through all the surrounding Galilean countryside
I don’t know what connection the “lectioneers” saw in these passages. For me it’s the issue of “authority”. That’s spelled out in the second reading: the crowd is astonished by the “first-hand-ness” of Jesus’ teaching. He spoke of God directly without claiming the authority of the Torah (Law) or the tradition of famous rabbis. There was a scandalous directness in his speech. This impression is confirmed by the fact that it is also effective speech. The super-personal power, generated by the uncleanness taboo in Jesus’ society, which has destroyed the man’s mind, “hears and obeys” the voice of the one who issues God’s command. It may seem wrong to compare the authority of Jesus with that of Hannah, yet her boldness prevails with God. She also expresses a direct faith in God, except that in her case, she gives God a good telling-off. If the Lord of all would just open his eyes to her humiliation and pay attention to her cause, then she would have a child and he would have a new servant.
No wonder Eli thinks she’s drunk: she talks to God as if He were there in front of her, but nothing is audible to him. She is equally blunt with the priest, telling him, as if he should have known it, that she was “pouring out her heart.” Eli recognises the authority of real experience, and gives her his blessing.
Neither Jesus nor Hannah made a fuss about their faith or claimed special inspiration. Nevertheless, their authority was clear.

To me the connection between the two is the importance of prayer and the Word of God in the life of the Christian in order to grow in faith.
Isn’t it wonderful that we can come before God in boldness and talk to him about anything in our life?
It is wonderful. Even more wonderful that God listens.