If I understand your question correctly, you could put the code in a function, to which you could pass the name you need as an argument and use that variable to construct your search strings.
for example:

def func(name_to_find):
# some code
    for answer in soup.select('p:contains("Question-and-Answer Session") ~ strong:contains("{n}") + p'.format(n=name_to_find)):
# some other code

and call it like so:

func('Dror Ben Asher')

In my (downloaded) HTMLs i have in the top of every file executives mentioned (like Dror Ben Asher” in the code below):

<DIV id=article_participants class="content_part hid">
<P>Redhill Biopharma Ltd. (NASDAQ:<A title="" href="http://seekingalpha.com/symbol/rdhl" symbolSlug="RDHL">RDHL</A>)</P>
<P>Q4 2014 <SPAN class=transcript-search-span style="BACKGROUND-COLOR: yellow">Earnings</SPAN> Conference <SPAN class=transcript-search-span style="BACKGROUND-COLOR: #f38686">Call</SPAN></P>
<P>February 26, 2015 9:00 AM ET</P>
<P><STRONG>Executives</STRONG></P>
<P>Dror Ben Asher - CEO</P>
<P>Ori Shilo - Deputy CEO, Finance and Operations</P>
<P>Guy Goldberg - Chief Business Officer</P>

Further along the html these executives name reaccurs multiple times where after the name follows an text element i want to parse
Example

<P>
<STRONG> Dror Ben Asher </STRONG>
</P>
<P>Yeah, in terms of production in first quarter, we’re going to be lower than we had forecasted mainly due to our grade.  We’ve had a couple of higher grade stopes in our Seabee complex that we’ve had some significant problems in terms of ground failures and dilution effects.  In addition, not helping out, we’ve had some equipment downtime on some of our smaller silt development, so the combination of those two issues are affecting us.
</p>

For now i have a code (see below) which identifies one executive “Dror Ben Asher” and graps all the text which accurs after in the P element. But I would like this to work for all executives and for Multiple html files where different executives are mentioned (different company).

import textwrap
import os
from bs4 import BeautifulSoup

directory ='C:/Research syntheses - Meta analysis/SeekingAlpha/out'
for filename in os.listdir(directory):
    if filename.endswith('.html'):
        fname = os.path.join(directory,filename)
        with open(fname, 'r') as f:
            soup = BeautifulSoup(f.read(),'html.parser')

print('{:<30} {:<70}'.format('Name', 'Answer'))
print('-' * 101)
for answer in soup.select('p:contains("Question-and-Answer Session") ~ strong:contains("Dror Ben Asher") + p'):
    txt = answer.get_text(strip=True)

    s = answer.find_next_sibling()
    while s:
        if s.name == 'strong' or s.find('strong'):
            break
        if s.name == 'p':
            txt += ' ' + s.get_text(strip=True)
        s = s.find_next_sibling()

    txt = ('n' + ' '*31).join(textwrap.wrap(txt))

    print('{:<30} {:<70}'.format('Dror Ben Asher - CEO', txt), file=open("output.txt", "a")

Does anyone have a suggestion to tackle this challenge?