If you go to a solicitor, and the solicitor wants to instruct a barrister on your behalf, then you are the lay client, you are the one who is paying for the service; you are perfectly entitled to say I want X or Y to be my barrister.
Now, the solicitor will normally from his own professional experience know which barristers are good, which aren’t so good, or which are good at certain matters, or which
aren’t so good at certain matters. And he will certainly want to instruct somebody he knows. And he will probably want to instruct somebody who is in a chambers where he already has an existing relationship.
It’s a matter for you whether you want to accept the advice of that solicitor to instruct a barrister he is suggesting, or whether you want to select your own barrister. That is an entire matter for you, but you are certainly entitled to select your own barrister and say to the solicitor, “ Look, that barrister covers this area of law.” for example, “ I heard he is very good, my neighbour had him.” or some such other recommendation, “therefore, I would like you to instruct that particular barrister.” And the solicitor has to do what you say, if you stick to your guns.